MODULE-4The value of Coefficient of contraction varies slightly with the available head of the...
Transcript of MODULE-4The value of Coefficient of contraction varies slightly with the available head of the...
Department of Civil Engineering, ATMECEDepartment of Civil Engineering, ATMECE
MODULE-4
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ORIFICES AND MOUTHPIECES
Orifice: An opening, in a vessel, through which the liquid flows out is known as orifice. This hole or
opening is called an orifice, so long as the level of the liquid on the upstream side is above the top of
the orifice.
The typical purpose of an orifice is the measurement of discharge. An orifice may be provided in the
vertical side of a vessel or in the base. But the former one is more common.
Types of Orifice
Orifices can be of different types depending upon their size, shape, and nature of discharge. But
the following are important from the subject point of view.
• According to size:
– Small orifice
– Large orifice
According to shape:
– Circular orifice
– Rectangular orifice
– Triangular orifice
According to shape of edge:
– Sharp-edged
– Bell-mouthed
According to nature of discharge:
– Discharging free Orifice
– Fully submerged Orifice
– Partially submerged Orifice
Venacontracta
Consider an orifice is fitted with a tank. The liquid particles, in order to flow out through the
orifice, move towards the orifice from all directions. A few of the particles first move downward, then
take a turn to enter into the orifice and then finally flow through it.
The liquid flowing through the orifice forms a jet of liquid whose area of cross-section is less then
that of the orifice. the area of jet of fluid goes on decreasing and at section C-C, the area is minimum.
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This section is approximately at a distance of half of diameter of the orifice . At this section, the
streamlines are straight and parallel to the each other and perpendicular to the plane of the orifice.
This section is called vena-contracta. Beyond this section, the jet diverges and is attracted in the
downward direction by the gravity.
Venacontracta
Hydraulic Coefficients
The following four coefficients are known as hydraulic coefficients or orifice coefficients.
1. Coefficient of Contraction: The ratio of the area of the jet, at vena-contracta, to the area of the
orifice is known as coefficient of contraction.
Cc =area o f the jet at venacontracta
area o f the ori f ice
The value of Coefficient of contraction varies slightly with the available head of the liquid, size
and shape of the orifice. The average value of Cc is 0.64
2. Coefficient of Velocity: The ratio of actual velocity of the jet, at Vena-contracta, to the theoret-
ical velocity is known as coefficient of velocity.The theoretical velocity of jet at Vena-contracta
is given by the relation, v =√
2gh
Cv =actual velocity o f the jet at Venacontracta
theoretical velocity
The difference between the velocities is due to friction of the orifice. The value of Coefficient
of velocity varies slightly with the different shapes of the edges of the orifice.This value is very
small for sharp-edged orifices. For a sharp edged orifice, the value of Cv increases with the head
of water. theoretical
3. Coefficient of Discharge The ratio of a actual discharge through an orifice to the theoretical
discharge is known as coefficient of discharge. Mathematically coefficient of discharge,
Cd =actual discharge
theoretical discharge=
actual velocity×actual areatheoretical velocity× theoretical area
=Cv×Cc
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Thus the value of coefficient of discharge varies with the values of Cc and Cv. An average of
coefficient of discharge varies from 0.60 to 0.64.
Discharge through the Orifice
Consider a tank fitted with circular orifice in one of its sides as shown in fig. let H be the head of
the liquid above the liquid above the center of orifice.
Flow through orifice
Consider two points 1 and 2 as shown in the fig. point 1 is inside the tank ans point 2 at vena-
contracta. let the flow be steady and at a constant head H.
Applying Bernoulli’s equation between 1 and 2
p1
ρg+
v21
2g+ z1 =
p2
ρg+
v22
2g+ z2
But, z1=z2,
Hence,
p1
ρg+
v21
2g=
p2
ρg+
v22
2g
Also
p1
ρg= H
p2
ρg= 0(atmospheric pressure)
v1is very small in comparision to v2 as area of the tank is very large compared to area of the jet
∴ H +0 = 0+v2
22g
∴ v2 =√
2gH
This is the expression for the theoretical velocity.
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Experimental determination of hydraulic coefficients
Determination of Cd
Water is allowed to flow through a orifice fitted to a tank under a constant head, H as shown in Fig.
The water collected in the measuring tank for a known time, t. the height of water in measuring tank
Value of Cv
is noted down. then actual discharge through orifice,
Q =area o f measuring tank×height o f water in measuring tank
Time(t)
and theoretical discharge = area of the orifice ×√
2gH
Cd =Q
a×√
2gH
Determination of Cv
Co-efficient of velocity,
Cv =x√4yH
where, x = Horizontal distance traveled by the particle in time ’t’
y= vertical distance between p and C-C (refer fig: Value of Cv)
Determination of Cc
Cd =Cv×Cc
Mouthpiece
Mouthpiece is a short tube of length not more than two or three times its diameter, provided in a
tank or a vessel containing fluid such that it is an extension of the orifice and through which also the
fluid may discharge. Both orifice and mouthpiece are usually used for measuring the rate of flow.
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Classification of Mouthpiece
1. Based on the position with respect to the tank or vessel to which they are fitted
(a) External mouthpiece
(b) Internal mouthpiece
2. Based on the shape:
(a) Cylindrical mouthpiece
(b) Convergent mouthpiece
(c) Convergent-divergent mouthpiece
3. Based on the nature of discharge at outlet of mouthpiece.
(a) Mouthpieces running full
(b) Mouthpieces running free
Flow through Mouthpiece
Flow through mouthpiece
Consider a tank having as external cylindrical mouthpiece of C/S area a1, attached to one of its
sides as shown in Fig. the jet off liquid enntering the mouthpiece contracts to form a vena-contracta
at a section C-C. Beyond this section, the jet again expands and fill the mouthpiece completely.
Let H=Heigh of liquid above the centre of mouthpiece
vc=Velocity of liquid at C-C section
ac=Area of flow at vena-contracta
v1=Velocity of liquid at outlet
a1=Area of mouthpiece at vena-contracta
Cc=Co-efficient of contraction.
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Applying continuity equation at C-C and (1)-(1), we get
ac× vc = a1× v1
∴ vc =a1v1
a1=
v1
ac/a1
But
ac
a1=Cc =Co− e f f icient o f contraction
taking Cd=0.62, we get aca1
= 0.62
∴ vc =v1
0.62
the jet of liquid from section C-C suddenly enlarges at section (1)-(1). Due to sudden enlargement,
there wiil be loss of head, h∗L which is given as
h∗L =(vc− v1)
2
2g
But,
vc =v1
0.62=
(v1
0.62 − v1
)2g
=v2
12g
[ 10.62
−1]=
0.375v21
2gApplying Bernoulli’s equation to point A (1)-(1)
p1
ρg+
v21
2g+ zA =
p2
ρg+
v22
2g+ z1 +hL
wherezA=z1 ,vA is negligible,
p1
ρg= atmospheric pressure = 0
∴ H +0 = 0+v2
12g
+0.375v2
12g
H = 1.375v2
12g
v1 =
√2gH1.375
= 0.855√
2gH
Theoretical velocity of liquid at outlet is vth=√
2gH
∴
Co-effficinet of velocity for mouthpiece
Cv =Actual velocity
T heoritical velocity=
0.855√
2gH√2gH
= 0.855
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Ccfor mouthpiece =1 as the area of the area of jet of liquid at out let is equal to area of the mouthpiece.
Thus,
Cd =Cv×Cc = 1.0×0.855 = 0.855
Borda’s mouthpiece
A short cylindrical tube attached to an orifice in such away that the tube projects inwardly to a
tank, is called as Borda’s mouthpiece or Re-entrant mouthpiece or internal mouthpiece.
Borda’s Mouthpiece running free
If the length of the tube is equal to its diameter, the jet of the liquid comes out from mouthpiece
without touching the sides of tube.
Mouthpiece running free
Discharage,
Q = 0.5×a√
2gH
where H= height of the fluid above the mouthpiece,
a=area of the mouthpiece
Boarda’s Mouthpiece running full
If the length of the tube is about 3 times its diameter, the jet comes out with its diameter equal to
diameter of the mouthpiece at outlet.
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Mouthpiece running full
Discharage,
Q = 0.707×a√
2gH
where H= height of the fluid above the mouthpiece,
a=area of the mouthpiece
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NOTCHES AND WEIRS
A notch is a device used for measuring the rate of flow of a liquid through a small channel or tank.
It is an opening in the side of a measuring tank or reservoir such that the water level is always below
the top edge of the opening.
A weir is a concrete or a masonry structure , placed in an open channel over which the flow occurs .
It is generally in the form of vertical wall, with Bell mouthed edge.
(a)Notch (b)Weir
Note:The sheet of water flowing over a notch or a weir is known as Nappe.
The bottom edge of the opening is known as ‘Sill’ or Crest.
Classification of notches
• According to the shape of opening
– Rectangular notch
– Triangular notch
– Trapezoidal notch
– Stepped notch
• According to the effect of the sides on the nappe:
– Notch with end contraction
– Notch without end contraction or suppressed notch
Discharge over a Rectangular notch
Consider a sharp edge rectangular notch with crest horizontal and normal to direction of flow. Let,
H= Head of water over the crest,
L=length of notch or weir
consider an elementary horizontal strip of water of thickness dh’ and length L at a depth ’h’ from free
surface of water as shown in Fig.
The area of the strip = L×dh
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Rectangular Notch
and theoretical velocity of water flowing through strip =√
2gh
The discharge dQ, through strip is
dQ =Cd×Areao f strip× theoriticalvelocity =Cd×L×dh×√
2gh
where Cd=Co-efficient of discharge.
The total discharge , Q, for the whole notch or weir is determined by integrating the above equation
between limits 0 and H.
Q =∫ H
0Cd L
√2gh dh =Cd×L×
√2g∫ H
0h1/2 dh
=Cd×L×√
2g[h1/2+1
12 +1
]H
0
=Cd×L×√
2g[h3/2
12
]H
0
=23
CdL√
2g[H]3/2
Discharge over a Triangular notch
Discharge over a triangular notch or weir is same. Let H= head of the water above the V-notch
θ =angle of notch
consider a horizontal strip of water of thickness ’dh’ at a depth of ’h’ from free surface as shown in
fig. from the fig we have
tanθ/2 =ACOC
=AC
H−h∴ AC = (H−h) tanθ/2
Width o f strip =AB = 2AC = 2(H−h) tanθ/2
∴ area o f the strip = 2(H−h) tanθ
2×dh
The theroritical velocity of water through strip =√
2gh
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Triangular Notch
∴ Discharge, dQ, through the strip is
dQ =Cd×area o f strip×Velocity(theoritical)
=Cd×2(H−h) tanθ
2×dh×
√2gh
∴ Total discharge, Q is
Q =∫ H
02Cd(H−h) tan
θ
2×√
2gh×dh
= 2Cd tanθ
2×√
2g∫ H
0(Hh1/2−h3/2)dh
= 2Cd tanθ
2
√2g[Hh3/2
3/2− h5/2
5/2
]H
0
= 2Cd tanθ
2
√2g[2
3H5/2− 2
5H5/2
]=
815
Cd tanθ
2
√2g×H5/2
Advantages of triangular notch/weir or rectangular notch/weir
• Expression for discharge for a right angled V-notch or weir is very simple.
• In case of triangular notch, only one reading, (H) is required for computation of discharge.
• Ventilation of triangular notch is not necessary.
• For measuring low discharge, a triangular notch gives more accurate results than a rectangular
notch.
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Discharge over a Trapezoidal notch
Trapezoidal notch is combination of rectangular and triangular notch or weir. thus the total dis-
charge will be equal to the sum of discharge through a rectangular notch and discharge through trian-
gular notch as shown in fig below
Trapezoidal Notch
Let H = head of the water above the V-notch
L = Length of the crest of the notch
Cd1= co-efficient of discharge for rectangular portion ABCD
Cd2= co-efficient of discharge for triangular portion [FAD and BCE]
The discharge through rectangular portion ABCD is given by
Q1 =23
Cd1×L√
2g×H3/2
The discharge through two triangular notches FCA and BCE is equal to discharge single triangular
notch of angle θ and it is given by equation as
Q2 =8
15Cd2 tan
θ
2
√2g×H5/2
∴ Discharge through tarpezoidal notch,
Q =23
Cd1×L√
2g×H3/2 +815
Cd2 tanθ
2
√2g×H5/2
Cipolletti Notch or Weir
Cipolletti weir is trpezoidal weir, which has sides slopes if 1 horizontal to 4 vertical as shown in
figure. Thus from fig,
tanθ
2=
ABBC
=H/2H
=14
∴θ
2= tan−1 1
4= 14o2
′
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Cipolletti weir
By giving this slopes to the sides an increase in discharge through the triangular portions ABC and
DEF of the weir is obtained. if this slope is not provided the weir would be a rectangular one, and due
to end contraction, the discharge would decrease. thus in case of Cippolletti weir, the factor of end
contraction is not required which is shown below. the discharge through a rectangular weir with two
end contraction is
Q =23×Cd× (L−0.2H)×
√2g×H3/2
=23×Cd×
√2g×H3/2− 2
15×Cd×
√2g×H5/2
Thus due to end contraction, the discharge decreases by 215Cd×
√2g×H5/2. This decrease discharge
can be compensated by giving such slope to the sides that the discharge through two triangular por-
tions is equal to 215Cd×
√2g×H5/2. Let the slope is given by θ/2. the discharge through a V-notch
of angle θ is given by
=8
15×Cd×
√2g× tan
θ
2H5/2
Thus
815×Cd×
√2g× tan
θ
2H5/2 =
215×Cd×
√2g× tan
θ
2H5/2
∴ tanθ
2=
215× 15
8=
14
or
θ/2 = 14o2′
Thus discharge through Cipoletti weir is
Q =23×Cd×L×
√2g×H3/2
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If the velocity of approach is considered,
Q =23×Cd× (L)×
√2g[(H +ha)
3/2−h3/2a
]Velocity of Approach
It is defined as the velocity with which the flow approaches/reaches the notch/weir before it flows
past it. The velocity of approach for any horizontal element across the notch depends only on its depth
below the free surface.
In most of the cases such as flow over a notch/weir in the side of the reservoir, the velocity of
approach may be neglected. But, for the notch/weir placed at the end of the narrow channel, the
velocity of approach to the weir will be substantial and the head producing the flow will be increased
by the kinetic energy of the approaching liquid.
Thus, if vais the velocity of approach, then the additional head ha due to velocity of approach,
acts on the water flowing over the notch or weir. So, the initial and final height of water over the
notch/weir will be (H +ha) and ha respectively. It may be determined by finding the discharge over
the notch/weir neglecting the velocity of approach i.e.
va =QA
where ’Q’ is the discharge over the notch/weir and ’A’ is the cross-sectional area of channel on the
upstream side of the weir/notch. Additional head corresponding to the velocity of approach will be,
Ha =v2
a2g
Example:- The discharge over a rectangular notch/weir of width L,
Q =23×Cd× (L)×
√2g[(H +ha)
3/2−h3/2a
]Broad crested weir
A weir having a wide crest is known as broad crested weir. Broad-crested weirs differ from thin-
plate and narrow-crested weirs by the fact that different flow pattern is developed.
Broad crested weir
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Condition for a weir to be broad or narrow.Let H= height of water, above the crest,
L=length of crest.
• If 2L > H, the weir is called broad crested weir.
• If 2L < H, the weir is called narrow crested weir.
(Refer above fig.)
Let h= head of water at the middle o weir which is constant
v=velocity of flow over the weir applying bernoullies equation to the still water surface on U/S side
and running water at the end of the weir,
0+0+H = 0+v2
2g+h
v2
2g= H−h
v =√
2g(H−h)
The discharge over weir
Q =Cd×Area o f f low× velocity
=Cd×L×h×√
2g(H−h)
=Cd×L×√
2g(Hh2−h3)
discharge is maximum if (Hh2−h3) is maximum or
ddh
(Hh2−h3) = 0 or h =23
H ∴
Qmax will be otained by substituting this valve of h in the above discharge equation
Qmax =Cd×L×√
2g[H×
(23
H)2−(2
3H)3]
=Cd×L×√
2g
√49
H3− 827
H3
=Cd×L×√
2g×0.3849×H(3/2)
= 1.705×Cd×L×H(3/2)
Submerged weir
When the water level on the down stream side of a weir is above the crest of the weir, then weir is
said to be submerged weir. Below fig shows a submerged weir.
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Submerged weir
The total discharge is obtained by dividing the weir into two parts. The portion between U/S and
D/S water surface may be treated as free weir and the portion between D/S water surface and crest of
weir as a drowned weir. Total disharge is given by
Q =23
Cd1×L×√
2g[H−h]3/2 +Cd2×L×h×√
2g(H−h)
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